Mass of rays on complete open surfaces
A connected, finitely connected, oriented, complete and noncompact Riemannian 2-manifold \(M\) without boundary is considered in the paper. Let \(M\) have \(k\) endpoints and let \(K\subset M\) be a compact set with the property that \(M\setminus \text{Int}\, K\) consists of \(K\) tubes \(\mathcal U_1,\ldots,\mathcal U_k\) such that each \(\mathcal U_ i\) is homeomorphic to \(\mathcal S^1\times [0,\infty)\) and that each \(\partial \mathcal U_i\) is a piecewise smooth simply closed curve. Let \(k(\partial \mathcal U_i)\) denote the curvature integral over the boundary curve \(\partial \mathcal U_i\) and \(c\) the total curvature, then the values \(s_i(M)=k(\partial \mathcal U_i)-c(\mathcal U_ i)\) are independent of the choice of the tubes. The relationship between these values and the ``mass of rays'' (i.e. measure \(\mathfrak M(A(p))\) where \(A(p)\) is the set of all unit vectors tangent to rays emanating from \(p\), \(\mathfrak M\) is the natural measure on \(\mathcal S^p(1)\subset T_pM\) induced from the Riemannian metric) is studied in the work. One of the main results is Theorem B. Assume that \(M\) admits total curvature \(c(M)\) and has \(K\) endpoints. Let \(\varphi\) be a simply closed smooth curve in \(M\) and let \[ B(t)=\{x\in M; d(x,\varphi)\leq t\},\quad S(t)=\{x\in M; d(x,\varphi)=t\}. \] If \(s_i(M)\leq 2\pi\) holds for each \(i=1,\ldots,k\) then \[ \lim_{t\to \infty}\Bigl(\int_{B(t)}\mathfrak M\circ A\, dM\Bigr)/\Bigl(\int_{B(t)}\,dM\Bigr)=\sum_{i=1}^K s^2_i(M)/(2\pi \chi(M)- c(M)), \] if \(2\pi \chi (M)-c(N)>0\); and \(\lim=0\) if \(2\pi \chi (M)-c(M)=0\), where \(\chi(M)\) is the Euler characteristic.
scientific article; zbMATH DE number 4209012